An adaptive least‐squares mixed finite element method for the Signorini problem. Issue 1 (25th July 2016)

• Record Type: Journal Article
• Authors: Krause, Rolf; Müller, Benjamin; Starke, Gerhard
• Is Part Of: Numerical methods for partial differential equations. Volume 33:Issue 1(2017:Jan.)
• Page Start: 276
• Page End: 289
• Publication Date: 2016-07-25
• Subjects: a posteriori error estimator -- contact boundary functional -- first‐order system least squares -- incompressible material -- Signorini contact problem; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22086
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract : We present and analyze a least squares formulation for contact problems in linear elasticity which employs both, displacements and stresses, as independent variables. As a consequence, we obtain stability and high accuracy of our discretization also in the incompressible limit. Moreover, our formulation gives rise to a reliable and efficient a posteriori error estimator. To incorporate the contact constraints, the first‐order system least squares functional is augmented by a contact boundary functional which implements the associated complementarity condition. The bilinear form related to the augmented functional is shown to be coercive and therefore constitutes an upper bound, up to a constant, for the error in displacements and stresses in H 1 ( Ω ) d × H ( div, Ω ) d . This implies the reliability of the functional to be used as an a posteriori error estimator in an adaptive framework. The efficiency of the use of the functional as an a posteriori error estimator is monitored by the local proportion of the boundary functional term with respect to the overall functional. Computational results using standard conforming linear finite elements for the displacement approximation combined with lowest‐order Raviart‐Thomas elements for the stress tensor show the effectiveness of our approach in an adaptive framework for two‐dimensional and three‐dimensional Hertzian contact problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 276–289, 2017